Math Final 109A

#1D Prove this conjecture: Even x Even= Even

(2m)(2n)=4mn=2(2mn) Even numbers can be expressed as 2m and 2n. Multiplying the two numbers of this form will yield 4mn or 2(2mn). Two times some number always yield an even, so this simplification proves my conjecture. ex: 2(2)=4 2(6)=12

#1F Prove this conjecture: odd (odd)=odd

(2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1 Two odd numbers can be represented as 2m+1 and 2n+1. Using the FOIL method we can simplify the equation to look like: 2(2mn+m+n)+1. Two times a number will yield an even as addressed in previous problems, but the additional one on the end results in an odd number. This proves the conjecture odd (odd)= odd ex: 3(3)=9 3(5)=15

#7 Justify operations with multiplication and division using decimals and fractions

** Fractions are building blocks of math*** MULTIPLICATION If you multiply .2 and .3 you receive an answer of .6. How is this possible? If you multiply using decimals you first multiply and ignore all decimals. Then put the decimal point in the answer - it will have as many decimal places as the two original numbers combined. If you look at multiplying decimals in the form of fractions .2 and .3 are the same as 2/10 and 3/10. When you multiply fractions you multiply across the top and across the bottom, so rather than 6/10, you receive an answer of 6/100 (because you multiply denominators 10 x 10). So .06 is equal to 6/100 Same idea if you multiply .02 and .03 and get an answer of .006. It is the same as multiplying 2/100 and 3/100. When you multiply fractions you multiply across top and bottom and you get 6/1000 or .006. DIVISION 6/.2 If the divisor (.2) is not a whole number, you must move the decimal place to the right to make it a whole number and then move the decimal place in the dividend the same number of places. Having a whole number divisor allows you to normally divide. So you would be dividing 60/2 and receive an answer of 30. To divide decimals as a fraction 6/.2, you first need to put 2/10 to create a fraction and then multiply by 10 in both the top and bottom to get rid of the fraction on the bottom of the original. To do this we cross out the tens in the bottom using the big one rule. Then we get rid of the 2/10 on the bottom and we have 60/2 which yields us 30. 6 x 10 -- 2 10 -- x -- *cross off tens* 10 1

#1E Prove this conjecture: even(odd)=even

2m(2n+1)=4mn+2m=2(mn+m) An even number can be expressed as 2m and an odd can be expressed as 2n+1. Knowing this, 2m (2n+1) would also be equal to 2(mn+m) when simplified using the distributive property. We know that anything times two yields an even so this proves the conjecture even (odd)=even ex: 2(3)=6 4(5)=20

#1C Prove this conjecture: odd+odd=even

2m+1+2n+1=2m+2n+2=2(m+n+1) An odd number can be looked at as an even number with a 1 added to it (2m+1). So, it you add two odds you are really adding an even to another even, then adding 1+1 which makes two. If you multiply any number by two you will result in an even number. This proves the conjecture. ex: 5+3=8 7+5=12

#1B Prove this conjecture: even+odd=odd

2m+2n+1= 2(m+n)+1 An even and an odd will result in an odd answer because 2(m+n) will always yield an even as addressed in the previous problem, but the additional one makes the result odd. ex: 2+3=5 4+7=11

#1A Prove this conjecture: even+even=even

2m+2n=2(m+n) Two even numbers being added can be written as 2m+2n. When you rearrange this terms you receive 2(n+m). Therefore any even plus another even will equal an even (you will always get number "x" multiplied by 2 making it even) ex: 2+2=4 4+6=10

#4B How to find prime

A prime number is defined as a number that has two distinct divisors, namely one and itself. Prime numbers are the building blocks of whole numbers and composites are composed of prime numbers. The Sieve of Eratosthenes helps us find prime numbers by crossing out all multiples and leaving us with numbers that are only divisible by one and itself ***Photo Below:

#9 Talk about percentage increase and decrease, discount and adding sales tax.

A scenario of percentage increase and decrease: INCREASE: There were only 200 endangered penguins left in Antartica. However, after a good year of food, the population grew by 5%. How many now? .05(200)=10 200+10=210 DECREASE: Your friend diets and goes from 145 pounds to 125 pounds. What was her percentage weight loss? 145-125=20 new-old 125-145 ----------= ---------= -.1379 old 145 She lowered her weight by 13% SALES TAX If I buy a phone for $100 and get a ten percent discount. How much is the product if the sales tax is 4.5%? 90% of original price(orig.) sale price(sales tax for NY) (.9)(100)= 90 90(.045)=4.05 90+4.05=**$94.05**

#4A How to find composite

Composite numbers are numbers that can be evenly divided by numbers other than one and itself Composites can be written as a product of prime such as 4=2 x 2= 2^2

#3B Approaching Division

Division is simply the inverse operation of multiplication. You approach long division as you would multiplication by splitting up what you are dividing by place value. So if you are dividing 32 into 469, you would first see how many times 32 goes into 4. If it can not, you use a placeholder zero, then move onto the next place. So rather than 32 goes into 4, you would be looking at how many times 32 goes into 46. You need to remember the rules that you can not divide by zero, otherwise you have an undefined number. Division requires 4 steps: Divide, Multiply, Subtract and Bring Down. You repeat this process for each place value until you are finished.

#10 Create a function machine, define function, use a real life function example and explain why it works

Function- for every input (x) we get one output (f(x) or y) output is dependent on input (independent variable) You can predict the output given your input in a function machine An example of a function machine would be a parking meter. If you have to pay a certain price to get a certain number of minutes to park the car. ex: one nickle- 3 min one dime- 6 min one quarter- 15 min This is a function because you expect a certain amount of minutes from each price. Function: Not a Function: f(x)= 2x+1 y^2=9 f(2)= 2(2)+1 y= +3/-3 f(2)=5 f(3)= 2(3)+1 x= [f(x)]^2 f(3)=7 *2 answers* How to determine a function: 1) using the vertical line test when graphing functions 2) When no element in x has two or more relationships

#2 Why is Hindu-Arabic the better system?

Hindu-Arabic is the system we currently use in mathematics. It is much more robust and the place value system is much less cumbersome than systems like the Egyptian system. The Egyptian system was also in base ten but lacked a place value system. All other systems were either in base 60 such as the Babylonian system or base 20 such as the Mayan system. The Hindu-Arabic system is a positional notation system, where everything is represented in base 10 (0-9) in decimal form. If you think of a simple addition problem such as 33 and 11 you receive an easy answer of 363 due to the Hindu Arabic system. However, if you tried to approach this problem using Roman numerals you would be adding XXXIII and XI. This instantly makes the problem more complicated. Now think about using long division with Roman numerals. The Hindu Arabic system makes a near impossible process seem easy. Zero is also used as a place holder and a number in the Hindu Arabic system. This makes it easier for students to redistribute when encountering a multiplication problem.

#8 Rational vs Irrational

Rational- a rational number is the ratio of two integers. They can be terminating and repeating numbers. Examples of Rational: 1/4=.25 (terminates) 1/3=.333333333333 (repeating) √9= 3 (terminates) 1/9=.1111111111111 (repeating) Irrational- umbers cannot be written as a ratio of two integers. They can be nonterminating and non repeating Examples of Irrational: π (non terminating) e √2= 1.41421356 (non repeating) √3=1.73205081 (non repeating) √6=2.44948974

#3 Approaching Multiplication

The key to a traditional algorithm is adding zero placeholders also known as the "magic number". When multiplying you must first line up the numbers, and then begin with the ones in the bottom number. So rather than multiplying the full number by the number on top you take the ones digit and multiply it by the number on top. Once finished you move to the tens place of the bottom number. This is where you need to add a 0 placeholder in the ones place in order to "hold the tens place". The third step would be to add the two numbers you received, the 1st when you multiplied by the ones place and the 2nd when you multiplied by the tens place. As long as you properly line up the answers and add your zero placeholder when multiplying by any place larger than the ones, you will end up with the correct answer. ex: 469 32 ----- 938 0 <--- placeholder for tens